Rohan Error Codes Correcting Gary Lecture 11 toolkit CMU Preliminaries Setting Error of Correcting Codes Linear Error Correcting codes and Hadamard Codes Hamming Reed Solomon Code
Definitions Basic Def An Error code is Correcting an injective map K length strings from n length strings to In Eric where will generally the alphabet We is take 0,1 2 binary l 8 g K is dimension message Message elements are messages in Block length bit string Msg n gk Code Codes this should Imsgl Ideally E Rate h Iblockl be close to 1
errors Et recover If I 2 Enc msg MSG running Msg Ei Eis Cei HammingDistanceimm Deff Hamming Dist number of positions at which the strings differ xi IYi31 DCx.ie Ei I ED I
d distance between any 2 vertices the minimum d Tiffany
Fact for each E the Unique decoding receiver there is a unique gets n she can recover is possible t E iff I
LINEAR CODE linear A code of length rank K and n linear is C with a subspace dimension K the where of vector Fgn space Fg is the finite field of q elements linear IF Deff code Ene Fgk Gx x where is vector a G is and a matrix G the is called Generator matrix full axle rank matrix lm G image of G which spans all linear combinations of rows
Notation linear code k d n q 9 b c length of h codeword not necessarily length of message K line distance D min III of Gx 2 z E.co wtz o.V Ct WE Fg et Ct is orthogonal to every vector in every C in vector codeword
n KII Ct code n k Enct Ff w to Htw Hgh maps H matrix Cn Kl xn is a Parity check matrix c Ct H of is rowspan 2 C C Hz 0 a D Defi Hamming weight wut Cw 0 W the least d Hamming wt Facts is of a c non Zero codeword
wth B y y y dlc a columns in min of number are linearly dependent which H 7roof ZEC d wt c min z to 2 min wtCZ Hz O 2 to
code Hamming 2 set binary q up i s I I I I 1 the columns span r ly is and matrix all possible binary strings of length Pot r full the identity µ is rank because it has matrix distance for 3 H
I Ham EE I 3 I r z let 2 n 3 n log ntl n A 9 block msg I Rate distance d errors 3 1 handle error n length Z string modified not HZ was Msg O else ei g i coordinate i for some e t 2 Msg
He H msg He Hz Il O n Perfect A perfect Code be code may mmmm balls in which the interpreted of one as the space radius t out fill exactly Bad distance rate Good A errors tolerated
Hadamard Code try I l l l l l l l l l Add zero's in row of if a for Hadamard Code Generator matrix
DEI Hadamard code Hadamard encoding of is defined the X of all inner as sequence with products x a x I at linear ne IFT define variate r Given polynomial IFI Az r xTa ai ni q L Mi's co efferienty ai's variables i mapping Like n to the of truth table
l4daDae i Ca.hu r I Facts Hadamard I is code code a r T t r block Msg distance let 2 n login In n code It distance errors log n M's rate block h
Super useful Reed Solomon codes Rs CRS code Def For 1 E ke h n G of symbols subset of Select cardinality n a Ist n SE Ag Eric Agh Fook M Mom message Ma I Pmk m ats Pm a C EGG mint x tmk mo Facts mum Linear Code
Eno mtm Ene EncCm m of polynomial adding coefficients Generator matrix k t each vow I a is a a for some AES Vandermonde matrix min dist Z K l h KH n Bad property In 8 KH n k n q these parameter for optimal I
Theorem Singleton Bond For k d a code n q K E 1964 h dtl mmmm I thanks
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